Question: Three positive integers have an arithmetic mean of 26 and median of 27. If the median is 5 less than the largest number, what is the smallest number?
Solution: The median of three integers is the middle integer. So the middle integer is $27$ and the largest integer is $27+5=32$. We also know that if the mean is $26$, then the sum of the three numbers is $26\times3=78$. We subtract the other two numbers to find that the third number is $78-27-32=\boxed{19}$.